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using gibbs equation to solve thermodynamic problems
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ME 291 – Thermodynamics I
The “Gibbs Equations” a.k.a. the “Tds Equations”
Tom Acker, PhD
Professor of Mechanical Engineering College of Engineering, Forestry and Natural Sciences
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^
^
^
Thermal Equilibrium
WarmObject
CoolObject
ThermalEquilibrium
Time
Both objects atsame temperature
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^
The
temperature difference drives the energy transfer as heat
between the objects ^ The
temperature is an indication of the internal energy
of
the objects The
temperature is related to the entropy
. The more
internal energy, the more ways to store the energy (morepossible macrostates and microstates,
w ), the more
entropy (
S = k ln w
^ S is maximized at thermal equilibrium
via the 2
nd^ Law for
an isolated system (microscopic: most likely macrostate) ^
WarmObject
CoolObject
ThermalEquilibrium
Time
T, S, & Thermal Equilibrium
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constant
2 1
^
B A^
U U U
U U
2 2
1 1
B A
B A^
B B A A^
2 1 2 1
0 E
^
) ,
neglect( 0
0
PE KE
U
E^
^ Note, the amount of energy on each side changes as heat istransferred between A & B. On a differential basis: Simplify with 1
Law:
B A^
B A^
dU dU
The change of energy in A andB are equal but opposite
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)
( )
( as
Rewrite
1 2
1 2
B B
A A^
U U
U U^
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B
A^
B B
A A
S U
S U
0
B
A^
B B
A A
This holds atthermal equilibrium
Plug (2) into (1):
^
A
B B A
A^ A
dU
S U
dU S U dS
B
A
^
^
B
A^
B B
A A A^
S U
S U
dU dS^
0
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v
v^
u T c^
c^ v^
is a property of a substance and is defined in terms of other properties (
u ,^ T ,and
v )
^
Compare to Specific Heat ^ Furthermore,
^ is a property that has the same value for twosubstances in thermal equilibrium; ^ since temperature is the same for two substances inthermal equilibrium, this new property must be related tothe temperature ^ Recall the definition of
c^ v
:
^ Similarly,
is a property of a substance
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B B A
A^ A
dU S U
dU S U dS
B
A^
B
A^
Recall Eq. (1):For any real process,
dS
0 (irreversible process),
And Eq. (2):Plug (2) into (1):
(^0)
^
B
A^
B B
A A A^
S U
S U
dU dS
Consider Any Process:
Let’s relate
dU
to the heat transfer between A & B A^
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W
Q
dU
^
Now write the first law for side A:
0 Heat transfer into A from B
Plug into (4):
^
B
A^
B B
A A A^
^
SideA (5)
Define Side A as the System
Q ^ A W ^ A
dS
is always positive If^ δ
QA
0, then the term in brackets must be > This occurs only when
TA
<^ TB
and
B
A^
B B
A A
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^ Since
when
;^ inverselyrelated
An appropriate relationship between
T ,^ S
, and
U^
is: Plug (6) into (4), and check sign on
dS
:
B
A^
B B
A A
B A^
T T^
S ^ U T^
1
T S^ U
1
or^
Thermodynamic Definition ofTemperature
B A A^
If^ T
, then B
δQ
0, and A^
dS
is +
If^ T
, then B
δQ
< 0, and A^
dS
is +
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^ For the units of Eq. (6) to be consistent,entropy must have units of energy pertemperature: SI^
Btulbm s BtuR S^
kJkg s kJK S^
Check Units on S
e temperatur
energy
Sof units
1 ^
e energytemperatur Sof units
English
So, our definition
makes sense
, is
dimensionally consistent
and satisfies the requirement that
S is maximized
at thermal
equilibrium
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^
Maximize Entropy
(^0)
B B BU
A AU A
B B B
A
A^ A
d S
d S
dU S U
dU S U dS
B
A
B
A
B A^
& ) , (^
^
US S From geometry:From 1
A
B^
A
B^
(^0)
^
B
A
B
A^
BU B
AU A A
B B
A A A
S
S
d
S U
S U
dU dS
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Recall Eq (6):
So this simplifies to
T S^ U
1
B
A^
U B B
U A A A
B A A
B A^
T T^
^
sincem
equilibriuat 0
B
A^
U B B
U A A A
Simplify
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^
Thermal equilibrium (
^ Mechanical equilibrium (
p^ A^
=^ p
B
A^
U B B
U A^ A
U S^
(^0)
B
A^
U B B
U A A A
d dS^ ^
^
^
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S
volume
e
temperatur energy
3 lengthe
temperatur
length force
e
temperatur
length force
2
e
pressuretemperatur
U S^
New Property:^ ^
P T
S
U
^
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B
A^
BU B
AU A A
d dS
B B A A A^
d dS
^
^
B A A^
d T dS
Check if it Makes Sense^ ^
^
^ If p
p^ B
, then
( p
p^ B
) < 0 and d
< 0 (volume of A shrinks)
and dS > 0 as it must be If p
p^ B
, then
( p
p^ B
) > 0 and d
> 0 (volume of A grows)
and dS > 0 as it must be
P T S U ^
What aboutsign of T ???
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It Makes Sense! ^ So, our definition^ ^
^ Next, let’s solve for P independent of T
P T
S
U
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(^1)
y x z^
z x y z x y
U S U S
S
U ^
S
^
T T P
U^
S
1 ) 1 (
or
S U
P^
(^1) (^)
S
Solve for the Pressure ^
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Interlude ^ We considered two isolated systems and cameup with relationships between
T
,^ S
, &
U
and
p ,
S , &
U
.
^ Simple compressible system ^ Relate changes: ^ Apply fundamental relations: 1
st^ law, 2
nd^
law
and some physical reasoning ^ Obtained:
) , (^
^
US S
d S dU S U
dS
U S
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